Function lupnt::RatioOfSectorToTriangleArea

Function Documentation

Real lupnt::RatioOfSectorToTriangleArea(const Vec3 &r1, const Vec3 &r2, Real tau)

Compute the ratio of the sector area to the triangle area.

Ratio eta of the area of the orbital sector swept between two position vectors to the area of the triangle they form with the focus, for a given time-of-flight tau (Gauss’s sector-triangle ratio, solved via Hansen’s method).

Used by conversions/state_conversions.cc (Gauss/Lambert-style position-velocity conversion, e.g. converting two position vectors plus a transfer time into a state vector) as part of the Herrick-Gibbs/Gauss orbit-determination calculation.

Parameters:

  • x – First vector

  • y – Second vector

  • tau – Time of flight O. Montenbruck and G. Eberhard, Satellite orbits: models, methods, and applications. Berlin : New York: Springer, 2000. doi: 10.1007/978-3-642-58351-3.

  • r1 – First position vector [any consistent length unit]

  • r2 – Second position vector [same unit as r1]

  • tau – Time of flight between r1 and r2, scaled per Montenbruck & Eberhard’s convention [consistent time unit]

Returns:

Sector-to-triangle area ratio eta; throws if Hansen’s iteration does not converge O. Montenbruck and G. Eberhard, Satellite orbits: models, methods, and applications. Berlin : New York: Springer, 2000. doi: 10.1007/978-3-642-58351-3.